Consider a natural exponential family parameterized by θ. It is well known that the standard conjugate prior on θ is characterized by a condition of posterior linearity for the expectation of the model mean parameter μ. Often, however, this family is not parameterized in terms of θ but rather in terms of a more usual parameter, such at the mean μ. The main question we address is: Under what conditions does a standard conjugate prior on μ induce a linear posterior expectation on μ itself? We prove that essentially this happens iff the exponential family has quadratic variance function. A consequence of this result is that the standard conjugate on μ coincides with the prior on μ induced by the standard conjugate on θ iff the variance function is quadratic. The rest of the article covers more specific issues related to conjugate priors for exponential families. In particular, we analyze the monotonicity of the expected posterior variance for μ with respect to the sample size and the hyperparameter "prior sample size" that appears in the conjugate distribution. Finally, we consider a situation in which a class of priors on θ, say Γ, is specified by some moment conditions. We revisit and extend previous results relating conjugate priors to Γ-least favorable distributions and Γ-minimax estimators.

Consider a natural exponential family parameterized by θ. It is well known that the standard conjugate prior on θ is characterized by a condition of posterior linearity for the expectation of the model mean parameter μ. Often, however, this family is not parameterized in terms of θ but rather in terms of a more usual parameter, such at the mean μ. The main question we address is: Under what conditions does a standard conjugate prior on μ induce a linear posterior expectation on μ itself? We prove that essentially this happens iff the exponential family has quadratic variance function. A consequence of this result is that the standard conjugate on μ coincides with the prior on μ induced by the standard conjugate on θ iff the variance function is quadratic. The rest of the article covers more specific issues related to conjugate priors for exponential families. In particular, we analyze the monotonicity of the expected posterior variance for μ with respect to the sample size and the hyperparameter "prior sample size" that appears in the conjugate distribution. Finally, we consider a situation in which a class of priors on θ, say Γ, is specified by some moment conditions. We revisit and extend previous results relating conjugate priors to Γ-least favorable distributions and Γ-minimax estimators.